Hecke algebra of ${\mathrm{GL}}_n$ over a 2-dimensional local field

Abstract

Using the $\mathbb{R}((X))$-measure, we define and study certain $\mathbb{C}((X))$-valued functions on ${\mathrm{GL}}_n(F)$ for $F$ a two-dimensional local field. In particular, we define a convolution product on such suitable functions, which leads us to define the Hecke algebra of ${\mathrm{GL}}_n(F)$. We then define the measurable $\mathbb{C}((X))$-representations of ${\mathrm{GL}}_n(F)$, and prove that function space is a candidate for such representations.